The denominator conjecture for SL₂-plethysm generating functions

Fix a positive integer $w$ and a partition $\mu \vdash w$. For any cell $c \in \mu$, denote the hook length of $c$ in the partition $\mu$ by $h_\mu(c)$. If $w$ is even, define

$$d_\mu(z,q) = \prod_{c \in \mu} (1-z^{h_\mu(c)}) \prod_{i=0}^{\frac{w}{2}-1} (1-q^{w-2i}z)$$

and if $w$ is odd, define

$$d_\mu(z,q) = \prod_{c \in \mu} (1-z^{2h_\mu(c)}) \prod_{i=0}^{\frac{w-1}{2}} (1-q^{w-2i}z).$$

Denominator conjecture. The expression

$$\frac{d_\mu(z,q) A_\mu(z,q)}{(1-z)^2}$$

is an element of $\mathbb{Z}[z,q]$.

This conjectural denominator is motivated by computations for $|\mu|\leqslant 10$ and is intended to improve the general rational denominator established earlier in the paper. Its resolution is not specified in the supplied text.